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## 5 Teaching Mathematics Through Problem Solving

Consider the following worthwhile-problem criteria developed by Lappan and Phillips (1998):

- The problem has important, useful mathematics embedded in it.
- The problem requires high-level thinking and problem solving.
- The problem contributes to the conceptual development of students.
- The problem creates an opportunity for the teacher to assess what his or her students are learning and where they are experiencing difficulty.
- The problem can be approached by students in multiple ways using different solution strategies.
- The problem has various solutions or allows different decisions or positions to be taken and defended.
- The problem encourages student engagement and discourse.
- The problem connects to other important mathematical ideas.
- The problem promotes the skillful use of mathematics.
- The problem provides an opportunity to practice important skills.

Key features of a good mathematics problem includes:

- It must begin where the students are mathematically.
- The feature of the problem must be the mathematics that students are to learn.
- It must require justifications and explanations for both answers and methods of solving.

## Mathematics Tasks and Activities that Promote Teaching through Problem Solving

## Choosing the Right Task

- Teachers must do the activity first. What is problematic about the activity? What will you need to do BEFORE the activity and AFTER the activity? Additionally, think how your students would do the activity.
- What mathematical ideas will the activity develop? Are there connections to other related mathematics topics, or other content areas?
- Can the activity accomplish your learning objective/goals?

## Low Floor High Ceiling Tasks

The strengths of using Low Floor High Ceiling Tasks:

- Allows students to show what they can do, not what they can’t.
- Provides differentiation to all students.
- Promotes a positive classroom environment.
- Advances a growth mindset in students
- Aligns with the Standards for Mathematical Practice

Examples of some Low Floor High Ceiling Tasks can be found at the following sites:

- YouCubed – under grades choose Low Floor High Ceiling
- NRICH Creating a Low Threshold High Ceiling Classroom
- Inside Mathematics Problems of the Month

## Math in 3-Acts

Act Three is the “reveal.” Students share their thinking as well as their solutions.

- Dan Meyer’s Three-Act Math Tasks
- Graham Fletcher3-Act Tasks ]
- Math in 3-Acts: Real World Math Problems to Make Math Contextual, Visual and Concrete

## Number Talks

- The teacher presents a problem for students to solve mentally.
- Provide adequate “ wait time .”
- The teacher calls on a students and asks, “What were you thinking?” and “Explain your thinking.”
- For each student who volunteers to share their strategy, write their thinking on the board. Make sure to accurately record their thinking; do not correct their responses.
- Invite students to question each other about their strategies, compare and contrast the strategies, and ask for clarification about strategies that are confusing.

## Saying “This is Easy”

When the teacher says, “this is easy,” students may think,

- “Everyone else understands and I don’t. I can’t do this!”
- Students may just give up and surrender the mathematics to their classmates.
- Students may shut down.

Instead, you and your students could say the following:

## Using “Worksheets”

- Provide your students a bridge between the concrete and abstract
- Serve as models that support students’ thinking
- Provide another representation
- Support student engagement
- Give students ownership of their own learning.

Adapted from “ The Top 5 Reasons for Using Manipulatives in the Classroom ”.

involves teaching a skill so that a student can later solve a story problem

when we teach students how to problem solve

teaching mathematics content through real contexts, problems, situations, and models

20 seconds to 2 minutes for students to make sense of questions

## Share This Book

## The Lesson Study Group

## Teaching Through Problem-solving

## What is Teaching Through Problem-Solving?

## Why Teaching Through Problem-Solving?

As students build their mathematical knowledge, they also:

- Learn to reason mathematically, using prior knowledge to build new ideas
- See the power of their explanations and carefully written work to spark insights for themselves and their classmates
- Expect mathematics to make sense
- Enjoy solving unfamiliar problems
- Experience mathematical discoveries that naturally deepen their perseverance

## Phases of a TTP Lesson

## WHAT STUDENTS DO

- Understand the problem and develop interest in solving it.
- Consider what they know that might help them solve the problem.

## WHAT TEACHERS DO

- Show several student journal reflections from the prior lesson.
- Pose a problem that students do not yet know how to solve.
- Interest students in the problem and in thinking about their own related knowledge.
- Independently try to solve the problem.
- Do not simply following the teacher’s solution example.
- Allow classmates to provide input after some independent thinking time.
- Circulate, using seating chart to note each student’s solution approach.
- Identify work to be presented and discussed at board.
- Ask individual questions to spark more thinking if some students finish quickly or don’t get started.
- Present and explain solution ideas at the board, are questioned by classmates and teacher. (2-3 students per lesson)
- Actively make sense of the presented work and draw out key mathematical points. (All students)
- Strategically select and sequence student presentations of work at the board, to build the new mathematics. (Incorrect approaches may be included.)
- Monitor student discussion: Are all students noticing the important mathematical ideas?
- Add teacher moves (questions, turn-and-talk, votes) as needed to build important mathematics.
- Consider what they learned and share their thoughts with class, to help formulate class summary of learning. Copy summary into journal.
- Write journal reflection on their own learning from the lesson.
- Write on the board a brief summary of what the class learned during the lesson, using student ideas and words where possible.
- Ask students to write in their journals about what they learned during the lesson.

## How Do Teachers Support Problem-solving?

## Additional Readings

## Can’t find a resource you need? Get in touch.

- What is Lesson Study?
- Why Lesson Study?
- Teacher Learning
- Content Resources
- Teaching Through Problem-solving (TTP)
- School-wide Lesson Study
- U.S. Networks
- International Networks

managing exam stress, education system, school management, administration, maths

## PROBLEM SOLVING METHOD: METHODS OF TEACHING MATHEMATICS

Problem solving is a set of events in which human beings was rules to achieve some goals – Gagne

Problem solving involves concept formation and discovery learning – Ausube

Steps in Problem Solving / Procedure for Problem solving

## 3. Formulating tentative hypothesis

Define union of two sets. If A={2,3,5}. B={3,5,6} And C={4,6,8,9}.

Step 1: Identifying and Defining the Problem

- The union of two sets A and B is the set, which contains all the members of a set A and all the members of a set B.
- The union of two set A and B is express as ‘AUB ’
- The common elements are taken only once in the union of two sets

Step 3 : Formulating Tentative Hypothesis

Thus on the basis of given data, the child will be able to solve the problem in the following manner

In the example it is given that

Step 5 : Verifying of the result

After testing and verifying his hypothesis the child will be able to conclude that

Thus the child generalises the results and apply his knowledge in new situations.

- This method is psychological and scientific in nature
- It helps in developing good study habits and reasoning powers.
- It helps to improve and apply knowledge and experience.
- This method stimulates thinking of the child
- It helps to develop the power of expression of the child.
- The child learns how to act in new situation.
- It develops group feeling while working together.
- Teachers become familiar with his pupils.
- It develops analytical, critical and generalization abilities of the child.
- This method helps in maintaining discipline in the class.
- This is not suitable for lower classes
- There is lack of suitable books and references for children.
- It is not economical. It is wastage of time and energy.
- Teachers find it difficult to cover the prescribed syllabus.
- To follow this method talented teacher are required.
- There is always doubt of drawing wrong conclusions.
- Mental activities are more emphasized as compared to physical activities.

Source: The Teaching of mathematics by KULBIR SINGH SIDHU (Sterling Publisher Pvt Ltd)

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## METHODS OF TEACHING MATHEMATICS

Friday, may 20, 2011, module 9: problem solving method.

## 3. Formulating tentative hypothesis

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## Problem Solving Strategies

## Pólya’s How to Solve It

- First, you have to understand the problem.
- After understanding, then make a plan.
- Carry out the plan.
- Look back on your work. How could it be better?

- What if the picture was different?
- What if the numbers were simpler?
- What if I just made up some numbers?

This brings us to the most important problem solving strategy of all:

## Problem 2 (Payback)

## Think/Pair/Share

Watch the solution only after you tried this strategy for yourself.

## Problem 3 (Squares on a Chess Board)

## Think / Pair / Share

- Describe all of the patterns you see in the table.
- Can you explain and justify any of the patterns you see? How can you be sure they will continue?
- What calculation would you do to find the total number of squares on a 100 × 100 chess board?

## Problem 4 (Broken Clock)

- What is the sum of all the numbers on the clock’s face?
- Can I find two consecutive numbers that give the correct sum? Or four consecutive numbers? Or some other amount?
- How do I know when I am done? When should I stop looking?

## Teaching Mathematics Through Problem Solving

By Tom McDougal and Akihiko Takahashi

## Figure 2 (Mathematics International, Grade 5, p. A93)

## Figure 3 (Mathematics International, Grade 5, p. A93)

## Figure 4 (Mathematics International, Grade 5, p. A94)

Idea 4: Divide: (area) ÷ (# of rabbits) = amount of area per rabbit

Idea 5: Divide: (# of rabbits) ÷ (area) = number of rabbits per unit area

He then invites a student to explain Idea 5: “I divided the other way…”

## Figure 5 (includes items from Mathematics International, Grade 5, pp. A93-94)

## Teaching & Learning: Creating a Culture of Academic Integrity

Step 2: Devise a plan (translate).

Step 3: Carry out the plan (solve).

Step 4: Look back (check and interpret).

Consecutive EVEN integers are even integers that follow one another in order.

Consecutive ODD integers are odd integers that follow one another in order.

Practice Problems 1a - 1g: Solve the word problem.

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## IMAGES

## VIDEO

## COMMENTS

The six steps of problem solving involve problem definition, problem analysis, developing possible solutions, selecting a solution, implementing the solution and evaluating the outcome. Problem solving models are used to address issues that...

The answer to any math problem depends on upon the question being asked. In most math problems, one needs to determine a missing variable. For instance, if a problem reads as 2+3 = , one needs to figure out what the number after the equals ...

In math, a computation method is used to find an answer in regards to any given problem. The most common computation methods make up the majority of basic math functions including addition, subtraction, multiplication and division.

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PROBLEM SOLVING METHOD. Maths is a subject of problem. · Steps in Problem Solving / Procedure for Problem solving. Identifying and defining the

The problem solving method is one, which involves the use of the process of problem solving or reflective thinking or reasoning. Problem solving

Pólya's How to Solve It · First, you have to understand the problem. · After understanding, then make a plan. · Carry out the plan. · Look back on your work. How

These are some possible questions the student could ask (Polya, 1957): Do I know what is the unknown? Do I know what are the data? Do I know

One mathematics teaching method that seems to be functioning in school is the use of open prob- lems (i.e., problem fields). Next we discuss the objectives of

A “teaching through problem solving” lesson would begin with the teacher setting up the context and introducing the problem. Students then work on the problem

What is the influence of teaching mathematics problem solving strategies on

Step 1: Understand the problem. · Step 2: Devise a plan (translate). · Step 3: Carry out the plan (solve). · Step 4: Look back (check and interpret)

developing the use of strategies in mathematical problem solving in classroom situations. 2. Background. 2.1 What is a strategy?